The goal of this Game is to delete the Minkowski-Einstein Theory of Relativity (“STR”). That theory asserts that the length and time deformations of the Lorentz Transformation Equations (“LTE”) don’t physically happen, that they are a consequence of “rotations” of the axes of moving co-ordinate systems followed by geometrical projections and hyperbolic trigonometry. The purpose of this end game is to prove to our physicists that this theory is incorrect; thus to get us to search for the underlying physical causes of the length and time deformations consequent to any change in the state of motion of physical systems; so as to begin to understand every structure and mechanism that exist in the Universe.

http://spinbitz.net/anpheon.org/html/Articles/endgame_new.htm

The Prior Moves

The very first move that led to STR was the ancient Greek philosphers’ notion that in order for matter to move there has to be an empty space into which some of it can move; thus that matter is made of ultimate particles (“atoms”) separated from one another by empty spaces. This doctrine became known as “the kinetic atomic theory”. That theory, however, is based on the hypothesis that matter itself is incompressible. (Since no one has ever found any such an incompressible material and since a space-filling compressible matter can easily be displaced by a moving body without any void spaces anywhere, this kinetic-atomic theory is, at best, open to question.)

The next move was the finding that light is a wave system, thus requires a conducting material substance that fills the spaces between atoms. Since a bit later it seemed that em waves are transverse, which requires a solid medium to conduct them, it was assumed that this medium (“the ether”) had to be some kind of generically different but also incompressible substance, even though atoms and thus “ponderable bodies” easily travel through it – which would be impossible, though that seems to have been ignored.

It was therefore accepted that an incompressible ether, which enacted and conducted em waves at a constant velocity c, filled the spaces between atoms of a ponderable body travelling at a velocity v through this stationary background ether. It was therefore thought that light waves must pass a moving aggregate, such as the planet Earth, at variable relative velocities in different local directions. Given that, then on the axis of motion a round-trip ray of light would move at an average relative velocity of c’ = Q = q2 = 1 – v2/c2 and at

c’ = q = sqrt(1 – v2/c2) along the perpendicular axes.

The next move was the failure of Michelson’s 1883 interferometer experiments to discover, from analysis of the expected interference patterns, the value of v – the “absolute velocity” of Earth, for which the universally stationary incompressible ether was taken as referent. No meaningful interference patterns were found.

Instead of calling into question the entire kinetic-atomic theory based on the hypothetical incompressibility of matter itself, other explanations of these “null” results appeared. One (around 1893), the Lorentz-Fitgerald contraction hypothesis, was that the arms of the apparatus contracted by q in the direction of Earth’s absolute motion. (The contractions were taken to mean that the incompressible ultimate particles of ponderable matter moved closer to each other in the intervening spaces, rather than that each of them bodily contracted.) Given that, then a ray moving at Q on the axis of motion would take 2q/Q = 2/q seconds for a round trip and a ray moving on a perpendicularly axis would take 2/q seconds, so the rays would round-trip both legs of the Michelson apparatus in the same period of time; thereby explaining the null result.

Around 1900, Larmour pointed out that there must also be a q-slowdown in the rates at which events happen in such q-contracted moving systems, including the rate at which their clocks beat, in order for the difference in the relative velocities of light, compared to c, to remain undetectable. With clock-rates running q slow, a ray traversing a q-contracted unit length on X would take q(2q/Q) = 2 seconds for a round trip and a ray on a perpendicularly axis would take q(2/q) = 2 seconds, whereupon the velocities in any direction at all, as measured by such physically deformed moving systems, would be equal to c, the velocity of light in the ether.

In 1904 Lorentz took the next step 1. He presented a set of equations that summarized all of the above. First he set forth Maxwell’s equations in terms of a classical euclidean-cartesian coordinate system taken as at rest in the ether. (A “euclidean-cartesian” system – “euclidean” for short – is one that has equal-sized units of measure in all directions and in all of which clocks have identical rates and settings everywhere.) Saying, “If the equations are at the same time referred to axes moving with the system, they become … “, he then wrote the equations that a euclidean system moving at v in the ether would plot for em events moving at c in that ether. In those equations, in which the equations have a different form than Maxwell’s because this moving system remained unchanged, the co-ordinates were still x,y,z and t. In principle, this was the same euclidean cs as before; now moving at an absolute velocity v. The stage for what became The Theory of Relativity had now been set.

The End Game

The first move in the end game itself was the very next step in Lorentz’s 1904 paper:

“§ 4. We shall further transform these formulae by a change of variables.

Putting

c2/(c2 – v2) = ß2, . . . . . (3)

and understanding by l another numerical quantity, to be determined further on,

I take as new independent variables

x’ = ßl x, y’ = l y, z’ = l z, . . . (4)

t’ = l t/ß – ßl vx/c2 . . . . . . . (5)”.

This was one of the most catastrophically misunderstood paragraphs in the history of science. Because he used the word “transform”, relativists later assumed that these were a somewhat defective version of the Lorentz Transformation Equations published a year later by Poincare’ 2. They think that there are two differently moving systems in eqs 4 and 5, and that (after working on the problem for over ten years) Lorentz – respected even by Einstein as perhaps the best mathematical physicists of the era – “forgot” to correct for the distance, vt, between the origins of the two differentaly moving systems at a time t. As a result of their misconception, the physical and mathematical meanings of eqs 4 and 5, and of the LTE as well, are generally misunderstood.

The following bordered portion, in which the quoted portions are the words of Lorentz, is from Appendix One of TPP 3. It explains what Lorentz’s eqs 4 and 5 actually mean.

“§ 8. Thus far we have used only the fundamental equations without any new assumptions. I shall now suppose that the electrons, which I take to be spheres of radius R in the state of rest, have their dimensions changed by the effect of a translation, the dimensions in the direction of motion becoming ßl times and those in perpendicular directions l times smaller.

“In this deformation, which may be represented by (1/ßl, 1/l, 1/l)

[which, for l = 1/q, becomes (Q,q,q) deformed in X,Y,Z] each element of volume is understood to preserve its charge.

“Our assumption amounts to saying that in an electrostatic system Z, moving with a velocity v, all electrons are flattened ellipsoids [*] with their smaller axes in the direction of motion. If now, in order to apply the theorem of § 6, we subject the system to the [reverse] deformation (ßl, l, l), we shall have again the spherical electrons of radius R. Hence, if we alter the relative positions of the centers of the electrons in [the moving system] by applying the deformations (ßl, l, l), and if, in the points thus obtained, we place the centers of electrons that remain at rest, we shall get a system, identical to the imaginary system Z’, of which we have spoken in § 6. The forces in this system and those in Z will bear to each other the relation expressed by [equation] (21).

“In the second place I shall suppose that the forces between uncharged particles, as well as those between such particles and electrons, are influenced by a translation in quite the same way as the electric forces in an electrostatic system. In other terms, whatever be the nature of the particles composing a ponderable body, so long as they do not move relatively to each other, we shall have between the forces acting in a system (Z’) without, and the same system (Z) with a translation, the relation specified in (21), if, as regards the relative position of the particles, Z’ is got from Z by the deformation (ßl, l, l ), or Z from Z’ by the deformation (1/ßl, 1/l, 1/l).”

Clearly, he was now treating one and the same physical system, “a [euclidean] system (Z’) without, and the same system (Z) with a translation”. It deforms one way when its velocity increases, and un-deforms the other way when it slows down. He was not doing transformations between co-ordinates of given events as plotted by coexisting differently moving coordinate systems. He was seeking to solve a very real physical problem: Why is there no interference pattern? Imaginary system Z’ was just an illustrative expedient with no ultimate role in his treatment of electromagnetic events as plotted by the moving system, the planet Earth.

“We see by this that, as soon as the resulting force is zero for a particle in Z’, the same must be true for the corresponding particle in Z. Consequently, if, neglecting the effects of molecular motion, we suppose each particle of a solid body to be in equilibrium under the action of the attractions and repulsions exerted by its neighbors, and if we take for granted that there is but one configuration of equilibrium, we may draw the conclusion that the system Z’, if the velocity v is imparted to it, will of itself change into the system Z. In other terms, the translation will produce the deformations (1/ßl, 1/l, 1/l).”

In short, if the component units comprising a larger unit change distances apart (in order to accommodate changed internal equilibrium conditions due to altered internal forces due to altered absolute velocity of the parent matter unit) the parent unit will, “of itself”, thereby have changed lengths and, because of the increased dinsity of the conducting medium, of rates of events. The Hafale-Keating 1970’s Pan Am flown atomic clocks experiment verified that this holds true for rates, within the single system, Earth. Sub-atomic nuclear physics attests that this holds true for everything else, hence lengths, as well.

Note, however, that no such experiments performed in a laboratory at rest on Earth have anything to do with displacements from points in Newtonian absolute space. The displacements are with respect to the material filling the bodily-enclosed laboratory itself. *

* Pressure-changes at the outer boundaries of a matter unit absolutely moving through the displaced variably-resistive material of the traversed field do permeate the unit and all its components. The “shock wave” has been noted, photographed, measured and cataloged for every planet in the Solar system. A stellar system and a galaxy or group of galaxies act similarly. So does a moving neutron. The laws of Nature are identical at every level of size. It’s just a matter of scale.

The next move was made by Poincare’ 2. Substituting x+vt for x in Eqs 4 and 5, and arbitrarily setting l equal to unity, he arrived at what he named “The Lorentz Transformation Equation Group”,

x’ = ß(x+vt), y’ = y, z’ = z and t’ = ß(t+vx/c2) . . . . the LTE.

This step entirely changed the meanings of Lorentz’s 1904 equations. Instead of denoting the deformations by which a moving euclidean coordinate system (a “cs”) internally deforms to become a non-euclidean relativistic system which then holds Maxwell’s equations applicable, it denotes the relations between co-ordinates plotted for given events by two differently moving systems; one or both of which are Lorentzian-deformed. In addition to changing the function – though not the physical content – of Lorentz’s deformation equations, Poincare’s step introduced several other complications:-

1. Adding rather than subtracting vt set system Z’ moving at -v relative to Lorentz’s euclidean system Z that was moving at +v. That requires that cs Z’ is at rest in the stationary ether – or in Einstein’s equally hypothetical empty space. (The mathematics is the same for both cases.)

2. It put the local time of successive clocks into the X-axis transforms, thus into the length of a unit rod of system A as plotted by esynched systems B. That hid the underlying fact that a unit rod of a deformed relativistic system is physically smaller in the direction of motion than a unit rod of a comoving euclidean system, as measured by both systems. That, in turn, hid the fact that despite each differently moving thus differently deformed esynched relativistic system measuring the other one’s unit rod as q-contracted, one of them is nevertheless physically shorter than the other. (The mechanism of the reciprocity of the transformations, and of how and why relativistic systems measure all differently moving other systems as q-contracted in the direction of relative motion, is minutely explained in Appendix 2 of The Painted Pony and won’t be done here.)

3. Poincare’s step not only changed the role of Lorentz’s 1904 equations, it seemed to eliminate any deformed coordinate system at all. Very deeply hidden, then, that made it seem as though the LTE hold good between differently moving euclidean systems; which they don’t! Rather, the LTE require and impose the very same underlying physical and mathematical deformations as eqs 4 and 5 from which they came, as explicitely set forth by Lorentz in his widely misunderstood 1904 paper. (This end game will show that if all differently moving systems remain undeformed euclidean systems, as present relativists seem to insist, Poincare’s LTE don’t apply. Only the galilean transformations hold good between differently moving undeformed euclidean systems.)

By inserting “vt” into the X axis transformation and vx/c2 into the time transformation Poincare’ had also set the stage for Minkowski’s later smearing of the difference between space and time themselves. Before that happened, however, Einstein tried to derive Poincare’s LTE without recourse to an ether medium filling space.

After presenting his method of setting clocks he wrote, in his 1905 paper 4,

“Let us in ‘stationary’ space take two systems of co-ordinates, i.e. two systems, each of three material lines, perpendicular to each other and issuing from a point. … Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring rod, and the clocks.

“To any system of values ksi,eta, zeta, tau, which completely defines the place and time of an event in the system k, there belongs a system of values x,y,z,t, determining that event relatively to the system K, and our task is now to find the system of equations connecting these quantities. …

“If we place x’ = x-vt, a point at rest in the system k must have a system of values x’,y,z, independent of time. We first define tau as a function of x’,y,z and t…. ”

As others pointed out, the latter stipulation is equivalent to appointing a third system, cs Z, attached to the moving system k but otherwise identical to Einstein’s system K that remained at rest “in ‘stationary’ space” thus to Poincare’s stationary euclidean system Z’. From the above discussion it follows that this attached system also remained euclidean. Einstein’s own usage requires that the relation between co-ordinates of the two euclidean systems, K and Z, are given by:

x’ = x-vt, y’ = y, z’ = z and t’ = t . . . . Equations 1.

Note, then, that these are NOT the Lorentz transformations! They’re the galilean ones; the only ones that apply between differently moving undeformed euclidean systems.

By later substituting x-vt for his x’ and using the symbol Ø(v) instead of l, he arrived at what I call the “Einstein Transformation Equations” (ETE). The ETE, which appear on page 46 of Reference 4, are:

“tau = Ø(v)ß(t – vx/c2),

ksi = Ø(v)ß(x – vt); eta = Ø(v)y; zeta = Ø(v)z,

where ß = 1/sqrt(1-v2/c2) and Ø is an as yet unknown function of v.”

They transform cs K co-ordinates into those cs k would plot for identical events. Note, then, that since relativistic transforms do not apply between euclidean systems, at least one of these – in this case cs k – is a physically and mathematically deformed thus non-euclidean system!

If we now reverse Einstein’s procedure by substituting x’ for x-vt in the ETA, we get,

tau = Ø(v)t/ß – ßØ(v)vx’/c2,

ksi = ßØ(v)x’, eta = Ø(v)y, zeta = Ø(v)z.

If we substitute l for Ø(v), these are Lorentz’s eqs 4 and 5! The very same physical deformations set forth by Lorentz are therefore mathematically built into the ETE and, through them, into the LTE of STR as well.

Despite the fact that their form is identical, there is an immense conceptual difference between the latter equations and Lorentz’s eqs 4 and 5. In Lorentz’s format, system K (Dr. Jekkyl) becomes deformed system k (Mr. Hyde). In essence, then, there was only one moving system in those equations. In Einstein’s format, however, system Z (Dr. Jekkyl) and system k (Jekkyl’s deformed brother, Dr. Hyde) coexist with each other, both of them moving at v relative to Poincare’s stationary system. Though the physical and mathematical deformations remain present, this conceptual difference will play a decisive role in this end game.

The next move was made by Minkowski. In a 1908 paper5 he said (page 76),

“Not to leave a yawning void anywhere, we will imagine that everywhere and everywhen there is something perceptible. To avoid saying ‘matter’ or ‘electricity’ I will use for this something the word ‘substance’.”

Matter, by any other name, remains the physical substrate enacting all local actions. In Minkowski’s terms it completely fills Newtonian space.

Whether it be called “matter” or “ether” or “substance” or “empty space” (with the properties only matter can possess), however, there is no such thing as an incompressible material. Indeed, the “curvature of the spacetime continuum” is a graphical way of describing the variable degrees of compression and consequent rates of events in different portions of resistively-compressible easily movable space-filling matter itself. That’s why GR mathematics fits physical reality so well.

After defining a few more of the relevant terms, Minkowski said (page 80),

“We will now introduce this fundamental axiom:-

The substance at any world-point may always, with the appropriate determination of space and time, be looked upon as at rest.”

His “appropriate determination of space and time” requires and imposes a coordinate system, itself taken as at rest in a local portion of this universal substance.

Given that portions of material are differently moving in different places, we can indeed attach a coordinate system to a given portion, which can then be regarded as the rest frame. The variable structure and actions of other portions of the wider material field can then be expressed in equations based on that rest frame.

Minkowski soon said (page 81),

“According to Lorentz any moving body must have undergone a contraction in the direction of its motion, and in fact with a velocity v, a contraction in the ratio 1: sqrt(1-v2/c2).

“This hypothesis sounds extremely fantastical, for the contraction is not to be looked upon as a consequence of resistances in the ether, or anything of that kind, but simply as a gift from above, as an accompanying circumstance of the circumstance of motion.”

Evidently Minkowski hadn’t read Lorentz’s paper, which set forth the force changes that cause moving physical systems (thus their attached coordinate systems) to bodily deform. He continued,

“I will now show by our figure that the Lorentzian hypothesis is completely equivalent to the new conception of space and time, which, indeed, makes the hypothesis much more intelligible. If for simplicity we disregard y and z, and imagine a world of one spatial dimension, then a parallel band, upright like the axis of t, and another inclining to the axis of t (see Fig. 1) represent, respectively, the career of a body at rest or in uniform motion, preserving in each case a constant spatial extent.”

It is implicit that a moving system that preserves a “constant spatial extent” does not physically deform. Instead, said Minkowsk, (pg 82):

“If OA’ is parallel to the second band, we can introduce t’ as the time and x’ as the space co-ordinate, and then the second body appears at rest, the first in uniform motion. We now assume that the first body, envisaged as at rest, has the length l, that is, the cross section PP of the first band on the axis of x is equal to l . OC, where OC denotes the unit of measure on the axis of x; and on the other hand, that the second body envisaged as at rest, has the same length l [my emphasis], which then means that the cross section Q’Q’ of the second band, measured parallel to the axis of x’, is equal to l . OC’. We now have in these two bodies images of two equal Lorentzian electrons, one at rest and one in uniform motion. But if we retain the original co-ordinates x,t, we must give as the extent of the second electron the cross section of its appropriate band parallel to the axis of x. Now since Q’Q’ = l . OC’; it is evident that QQ = l . OD’. If dx/dt for the second band is equal to v, an easy calculation gives

OD’ = OC sqrt(1-v2/c2),

therefore also

P : QQ = 1: sqrt(1-v2/c2).

But this is the meaning of Lorentz’s hypothesis of the contraction of electrons in motion.”

Minkowski did not provide the “easy calculation”. Sommerfeld did, in the “Notes”. Careful study of his “proof” shows that there is a mathematical error in it which, when seen and corrected, proves that Minkowski was wrong. Toward the end of his proof, Sommerfeld wrote (page 93),

“This, together with (4), gives the proportion, OD : OA’ = OD’ : OA, which, as

OA’ = OC’ and OA = OC,

is identical with OD : OC’ = OD’ : OC employed on page 82, line 29.”

In Minkowski’s Figure 1, however, the length of line OA isn’t equal to that of line OC! Therefore both the proof and Minkowski’s assertion were false. Indeed, the real reason Minkowski’s Figure 1 diagram worked is that he drew the moving electron physically shorter than the other one in advance!

Perhaps Minkowski noticed this discrepancy a year or so later, which may be why he switched to his hyperbolic rotation method instead. In it, the length of the moving electon on its own slanted X’ axis is drawn equal to that of the stationary one on its own X axis. The fact that it is nevertheless q-contracted is built into the angle of slant, which is a function of the value of v. Hence the physical deformation is masked by that mathematical procedure. So is the fact that, as shown above, the LTE do not hold good between identical euclidean systems both of which maintain the same spatial extent.

Meanwhile, Minkowski’s paper continued,

“If on the other hand we envisage the second electron as at rest, and therefore adopt the system of reference x’,t’, the length of the first must be denoted by the cross section P’P’ of its band parallel to OC’, and we should find the first electron in comparison with the second to be contracted in exactly the same proportion; for in the figure

P’P’ : Q’Q’ = OD : OC’ = OD’ : OC = QQ : PP.

“Lorentz called the t’ combination of x and t the local time of the electron in uniform motion, and applied a physical construction of this concept, for the better understanding of the hypothesis of contraction.

“But the credit of first recognizing clearly that the time of the one electron is just as good as that of the other, that is to say, that t and t’ are to be treated identically belongs to A. Einstein. [*] Thus time, as a concept unequivocally determined by phenomena, was first deposed from its high seat. Neither Einstein nor Lorentz made any attack on the concept of space [**], perhaps because in the above-mentioned special transformation, where the plane of x’,t’ coincides with the plane of x,t, an interpretation is possible by saying that the x-axis of space maintains its position.”

* In his 1905 paper, Einstein said, “Thence we conclude that a balance clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.” (Evidently Minkowski hadn’t read that paper either.)

** What Einstein was blocked from seeing by his own w(v) = Ø(v) = 1 covert denial of real changes in lengths and rates of moving physical systems is that, as Lorentz clearly and precisely pointed out replete with equations and verbal explanations, the “space” of each differently moving coordinate system must deform in exactly the same manner as the deformed physical system to which it is attached. What relativists are therefore blocked from seeing is that the “space” attached to a moving relativistic system is no longer isometric; it (as is its manifold of spacetime points) is q-contracted in the direction of motion!

Minkowski had now introduced the basis of what later became the relativistic thesis that the deformations of a moving system are exclusively due to geometrical projections. In order to check that thesis we will now put together Einstein’s format (co-travelling and co-existing Dr. Jekkyl and his brother Dr. Hyde), Lorentz’s deformation equations describing the deformations of Hyde compared to Jekkyl, Poincare’s LTE, and then Minkowski’s hyperbolic rotations.

In the following discussion cs K’ (ksi,eta,theta; tau) represents Poincare’s stationary euclidean system; cs K (x,y,z; t) represents Lorentz’s moving but still euclidean system (Dr. Jekkyl); and Einstein’s esynched cs k (x’,y’,z’; t’) represents Lorentz’s relativistically deformed co-moving system (Mr. Hyde who, though he became invisible in the Poincare’ format, will be perhaps the most powerful player of all).

Some of the concepts and terms we will now need are defined as follows (from ITB 6):

Physical space: Physical space is the totality of physical extension in all directions. It may also be called “absolute space” or “objective space ” herein. Any smaller portions of it will also be denoted by these terms.

Absolute motion is a change of place of anything moving in physical space.

Physical time: This denotes the totality of duration, past, present and future; and any portion or portions of it.

Physical relation: This denotes the juxtaposition between things in physical space and time, whether or not that relation is measured. It may also be called absolute relation.

Dimension: A dimension represents one appointed category of the many variables in the physical world. It is the name of the specified variable we choose to measure. Each dimension has a unit of measure, arbitrarily appointed and accepted by Man via a Convention of people judged qualified to do so. They choose some physical thing to represent such a unit, which nevertheless remains a conceptual abstraction invented by us.

Frame of reference: For measuring purposes, a specified identifiable object has to be chosen as the zero point from which measurements can begin. The chosen referent is the center of what is thereby an extended frame of reference. Each and all such frames of reference are imaginary, thus can freely move through each other.

Co-ordinate system: A co-ordinate system is a conceptual device quantifying the space attached to a specified frame of reference. The chosen referent is the zero point of such a system.

Metrical space: Metrical space denotes the continuum of quantified points belonging to the arbitrarily three-dimensional co-ordinate system permanently attached to a chosen referent. It is the dimension that measures arbitrarily selected places in objective space. Its unit is one meter, defined by Conventional agreement.

Although both metrical space and physical space are generally called simply space, the two are not identical. Metrical space measure things that exist in absolute space.

Metrical time: This is the dimension with which we measure arbitrarily selected intervals of duration. Its unit is one second, as beat off by a standard clock attached to a specified frame of reference. It is generally called simply “time”.

Here are the images of two meter rods, which are a different size than each other. Each has esynched clocks, @ and @’ respectively, marking the time at its point.

@______________________@

@’________________________________@’ Figure A.

Each meter rod will sometimes be called a “unit-rod”, below, the shorter one belonging to system k and the other to system K.

The next Figure illustrates the situation treated by Lorentz’s x’ = ßl x of eqs 4. Systems K and k are co-travelling at v and the origins and axes of K and k permanently coincide. In the figure, the X axis and the unit-rod of K are at the top and X’ and the coinciding unit-rod of k at the bottom of the shared axis-line. All points per axis of euclidean cs K remain where they were beforehand, while those in the direction of motion of thereby-deformed relativistic system k have moved closer together.

Figure B.

The physical lengths of the two meter-rods (their actual sizes) are different than each other. The unit-rod of cs K extends to x = 1, at the upper point in the diagram. The coinciding but physically shorter unit-rod of cs k extends to x’ = 1, at the indicated lower point.

Letting “proper length” denote the length of a rod as measured by its own coordinate system, the proper length of each rod is “1 meter”. As measured by each other, however, if we let l = 1 the unit-rod of k is q-shorter than the unit-rod of K.

All other systems will agree that the proper lengths of the k and K unit-rods are “1 meter long”; wherefore the “proper lengths” are “invariant”. All systems, however, including K and k, will agree that the k rod is shorter than the K rod, even though they are at rest relative to each other. Even so, all relativistic systems (other Misters Hyde) will plot any differently moving meter rod as q-contracted. (The fact that they will measure the others’ rods as q-contracted as a function of relative v, alone, is an artifact produced by the use of the offset “times” of their esynched clocks and their own differently deformed unit-rods, as measuring tools.)

There is thus a vast difference between “the physical length” of two meter rods compared to each other versus “the metrical length of two differently moving meter rods” as measured by each differently moving thus differently deformed internally esynched relativistic system. This illustrates the fact that although the “proper length” of a unit-rod, thus one meter as measured in its own system, is “invariant” (in the STR meaning of the word), the physical length is not! The physical length of a moving rod is a function of its absolute velocity wrt either Lorentz’s ether, Einstein’s empty space or Minkowski’s locally stationary “substance at any world-point “.

Having set forth the force changes that cause the physical deformations of sizes, Lorentz then presented the second thing that is required in order for eqs 4 and 5 to hold good. In section 10 he wrote,

“(a) Let A’1, A’2, A’3, etc., be the centers of the particles in the system without translation (Z’); neglecting molecular motions we shall assume these points to remain at rest. The system of points A1, A2, A3, etc., formed by the centers of the particles in the moving system, is obtained from A’1, A’2, A’3, etc., by means of a deformation (1/ßl, 1/l, 1/l). According to what has been said in # 8, the centers will of themselves take these positions A’1, A’2, A’3, etc., if originally, before there was a translation, they occupied the positions A1, A2, A3, etc.

“We may conceive any point P’ in the space of the system Z’ to be displaced by the above deformation, so that a definite point P of corresponds to it. For two corresponding points P’ and P we shall define corresponding instants, the one belonging to P’, the other belonging to P, by stating that the true time at the first instant is equal to the local time, as determined by (5) for the point P, at the [corresponding] instant. By corresponding times for two corresponding particles we shall understand times that may be said to correspond, if we fix our attention on the centers A’ and A of these particles.”

In addition to the deformations of the physical system itself, then, the metrical space and time per frame of reference and attached coordinate system must similarly deform, thus so must the thereby unique spacetime per differently moving system!

The next Figure illustrates this wrt an X,Y section. Cs K (thick lines) remains euclidean and the space of co-travelling cs k (thin lines, some overlaying those of K) has contracted by q in the direction of motion.

As shown on the left in the Figure, the grid belonging to relativistic cs k is q-shorter in the direction of motion than that of euclidean cs K. The Figure on the right shows how the metrical space, thus the spacetime of k, changes as v of K and k does.

In Lorentz’s eq 5, which (setting l = 1) becomes t’ = t/ß – ßvx/c2, the right side of the equation contains two different things. The first, t/ß, gives the time t’o of the origin clock of k, running at qt rates compared to t. The second, -ßvx/c2, gives the local offset of the k-clock at x’ = ßx/c, as a function of v/c, compared to the time of its own origin clock. The next Figure illustrates the meaning of the first of these two.

The top image in the Figure presents a Y,t cross-section because there are no offsets in the directions perpendicular to the direction of motion, here X,X’. The k-time runs q-slower than that of K, so at t = 1

t’ = qt = .8; and at t’ = 1, t = 1.25. The lower image opens Pandora’s box!

The deformations of size are attributable to the force-changes consequent to absolute velocities. The rate changes can be attributed to the resulting density changes of the given system. However, other than by hand-changing the settings per clock of system k, either arbitrarily or via Einstein’s esynching method, no local-time offsets are produced by absolute velocities alone. In the lower image, then, system k hasn’t done that yet; wherefore all clocks on X’ still register the same time as their own origin clock.

Although this agrees with the present relativistic assertion that “rotations” accompanied by geometric projections and hyp-trig “explain” the LTE even though nothing actually happens to differently moving euclidean systems, it rules out Lorentz’s eq 5; thus the Poincare’ time transformation obtained from it merely by substituting x+vt for x and gathering terms.

We will now analyze “Minkowski rotation” mathematics. The following figure presents an actual rotation of the X and X’ axes of systems K and k relative to Ksi of stationary system K’. The K rod has the same size as that of the stationary K’ rod; just as a Minkowski diagram requires. We must also include the X’ axis and unit-rod of cs k in that diagram. Where will they be?

Well, as plotted by K’, system k moves at the same velocity as system K. Presumably, then, the X and X’ axes are identically rotated thus coincide. We will therefore draw it on the same slanted line.

Figure C.

Figure C shows the identically rotated X and X’ axes of comoving K and k relative to Ksi of cs K’. The angle a of the rotation of the permanently coinciding X and X’ axes of the comoving systems is a = atan(c/{v/c}). For v = .6c, a = 59.036243 degrees, as implied in Figure C.

By a geometrical projection and circular trig we get,

ksi = cos(a)x = .5144958,

as shown in Figure C. If we projected a line from x = 1 to line tau, the trig would give us,

tau = sin(a)x = .8574929.

Doing the same thing wrt the k rod, we’d get ksi = .4116 and tau = .6859943.

None of these results fit the LTE. Even though no relativist suggested that they would, this shows that actual circular rotations followed by geometric projections and trigonometry don’t match LTE values.

We will now investigate hyperbolic rotations in “flat spacetime” (a 4-d manifold of points, each equidistant from its immediate neighbors in all directions) and the associated geometric projections and hyperbolic trig. In the following Figure, the page itself may be considered as a 2-d plane-slice of a flat spacetime continuum. Each point of the continuum, thus of the plane, is relativistically called an “event”. Since no coordinate system is an intrinsic part of the continuum, the events are independent of any x,y,z or t co-ordinates. In the figure, the letters a, b, c and d denote specific such events, each of which has a time and space parameter.

Figure 1.

It may be noted that the time parameter also exists in the other two spatial directions, thus may be considered as existing not only on the page, but perpendicular to it at every event-point as well.

Now, in Einstein’s words,

“Let us in ‘stationary’ space take two systems of co-ordinates, i.e. two systems, each of three material lines, perpendicular to each other and issuing from a point. Let the axis of X of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system be provided with a rigid measuring-rod and a number of clocks, and let the measuring-rods, and likewise all the clocks of the two systems be in all respects alike.”

Allow, then, that two coordinate systems, K (x,y,z;t) and k (x’,y’,z’;t’) are mutually at rest with their origins and respective axes coinciding. We will now draw the two systems on the spacetime page.

Figure 2.

The origins of the two systems are at spacetime event o, which is therefore at t = t’ = x = x’ = 0, and the X,X’ axes are through events a,b, c and d; each of which, and all others, may now be identified by x,t and x’t’ co-ordinates per point.

Einstein continued,

“Now to the origin of one of the systems (k) let a constant velocity [v = .6c] be imparted in the direction of the increasing X of the other system and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring rod, and the clocks.”

We will let above system K’ be the one that remains at rest in Einstein’s “‘stationary’ space” (represented by our slice of Minkowski’s euclidean spacetime continuum) and let systems K and k be co-traveling along the Ksi axis at v = .6c. In the next Figure, which includes the hyperbolically rotated X,X’ axes and their unit-rods, the Ksi,tau axes of system K’ therefore take the place of the X,X’ and t,t’ axes of Figure 2.

Figure 3.

We will assume that systems K and K’ remain identical and euclidean and cs k remains q-shrunken; that at tau = 0 all clocks register zero; and that k-clocks, either spontaneously or by hand-changes, run q-slow compared to those of K and K’.

Let each observer attached to cs k per successive point x’ now be told that at t’ = 0 each is to turn back his clock by -vx’/c2 seconds, in which x’/c is the numerical value of his k-point and v/c is the velocity as plotted by cs K’. Let cs k then check the “synchrony” of its reset clocks via Einstein’s defined method. It will find them correctly esynched! (This procedure is set forth only to avoid the complication that it would take a lot of time to perform esynching, and we want to begin our spacetime diagramming at tau=t=t’o=0.)

For v = .6c, the angle of rotation is a = arctanh(v/c) = .6931472 radians = 39.7144 degrees. By dropping lines perpendicular to Ksi from the endpoints of the two co-moving rods we obtain the geometrical projections of the rods at tau = 0. The ends of the undeformed K-rod will be at ksi = 0 and, as shown in Figure 3, to ksi = .8, so the length of that rod will be dksi = q and we thus have the numerical equivalent of a “Lorentz contraction”. The K’ to K transformations, however, aren’t the LTE, they’re the galilean ones! They yield x = ksi – vtau = dksi = 1! (They also yield t = tau for all values of tau.)

Doing the same projection wrt the q-shrunken k-rod we would find that dksi = .64 = Q. The Lorentz transformation, however, gives us x’ = ß(ksi – vtau) = .8; which requires that dksi = q!

If we now drew a geometrical projection of a K’ unit-rod to the X’ axis in the latter diagram, we’d find that instead of being q-contracted as the LTE require, the K’ rod would be the same length as the k rod, as so-determined by cs k. This fits only the case in which the moving system k is Q, q, q, 1 deformed. For that case, the value of phi(-v) in the inverse ETE is phi(-v) = 1/phi(v) = q and the applicable transform to ksi from x’ = 1 is ksi = ßphi(-v)(x’ + vt’) = 1. Accordingly, via geometrical projections the stationary euclidean system K’ would plot the q-shrunken k-rod as Q-contracted in the direction of motion and the deformed relativistic cs k would plot the K’ rod as being the same length as its own; neither of which fits the LTE.

Let euclidean cs K be moving along Ksi at v = .6c wrt a 2-d Ksi,Eta slice of the spacetime manifold. A hyperbolic rotation of the axes of system K followed by hyperbolic trigonometry gives us

x = cosh(r)ksi – sinh(r)eta; y = cosh(r)eta – sinh(r)ksi,

in which, as above, r = .6931472. At t = t’ = 0, when the origins coincide, let an event be plotted at ksi = 1, eta = 0. We then get,

x = cosh(r)ksi – sinh(r)eta = 1.25 – 0 = 1.25.

(It LOOKS like a lorentz transform, so far.) At that same instant, however, let another event be plotted at ksi = 0, eta = 1. From the above hyperbolic equation we then get,

y = cosh(r)eta – sinh(r)ksi = 1.25 – 0 = 1.25.

From the lorentz transform, however, we get y = eta = 1.

If we now applied the hyperbolic trig to the t axis of K for tau = ksi = 1, we’d get,

t = cosh(r)tau – sinh(r)ksi = (gamma tau) – gamma(v/c2)ksi = gamma(tau – vksi/c2) = .5,

which does fit the LTE. Accordingly, if actual hyperbolic rotations occur, the hyperbolic trig and geometric projections do fit the LTE in x and t but not in y and z.

Either way, actual circular or hyperbolic rotations and applicable geometrical projections and trigonometry not only fail to give the correct values for the galilean transformations between differently moving euclidean systems K and K’, they impose an entirely different set of transformations than the LTE when applied to the K’ to k case that does fit the LTE. The Minkowski method thus gives the “right” answer for the wrong physical case and the wrong answer for the physical cases the LTE do treat.

Accordingly, though deeply hidden within the mathematics itself, you can’t have both:- The physically undeformed rods and universally identical clock-rates and settings that Minkowski’s relativity-theory asserts; and the LTE. They don’t fit each other.

Therefore, in order to fit the LTE in all four axes, something entirely different than rotations and geometric projections is mathematically inserted by the hyperbolic trig itself. That “something else” is embedded within the latter equation, in which cosh(r) = gamma and sinh(r) = gamma(v/c). We will examine it below.

The Closing Moves

We will now plot the hyperbolic worldlines of k clocks at o, a, b, c and d at tau = 0, as tau increases to tau = 1. That gives us the worldpoint per k-clock at that time. A line through those points is where they and their points x’ and therefore all points x’ comprising the X’ axis are, thus where the X’ axis itself is, at tau = 1. As shown in the next Figure, it is on the ksi-axis-line through tau = 1, thus, as is the X axes, is perpendicular to tau and parallel to the Ksi axis.

Figure 4.

Although each and all points x’ do move at an angle wrt the Ksi axis in Figure 4, they do so in unison. Therefore, the X’ axis doesn’t rotate, it bodily moves at that radian-angle, a = atanh(v/c); always remaining parallel to the Ksi axis. Similarly, at any time tau the X and coinciding X’ axes are on or parallel to Ksi.

Minkowskians assert that, “The x-axis is the locus of events where y=0, z=0, and t=0. Similarly, the t-axis is the locus of events where x=y=z=0. And the x’-axis is the locus of events where y’=z’=t’=0, which is a completely different locus of points than the x-axis”. They then provide a hyperbolic Minkowski-rotation figure as follows, in which euclidean K takes the place of above cs K’ and inertially moving system k remains an undeformed euclidean system.

As shown above, however, the X’ axis isn’t actually rotated in any way at all. Accordingly, the rotations of the X’ and t’ axes in the standard Minkowski-rotation figures is unjustified by the pure mathematics itself. Therefore, so is the notion that the lorentzian deformations are “explained” as a consequence of geometrical projections based on the (mythical and mathematically denied) rotations of the axes of a relatively-moving euclidean coordinate system.

Line X’ therefore represents something entirely different than the X’ axis itself! As it now emerges from “Pandora’s box”, we will see what that really is.

Figure 5.

Since K-clocks have the same settings as K’ clocks and each other, it makes no difference where we place the perpendicular t axis. Since no two esynched clocks on X’ of the moving relativistic cs k have the same time as their origin clock or each other at the same instant, however, only the k-origin clock registers t’o at a given instant. Therefore the t’ axis is drawn through the k-origin and, as shown in Figure 5, moves with it; always remaining parallel to the tau and t axes and perpendicular to Ksi, X and X’.

In Figure 5, line X’ of the prior figure has now been labelled the t’x’ = t’o = 0 line. This isn’t the X’ axis line! As shown in the next figure, it’s the line that each sucessive esynched cs k clock will eventually reach when it later registers t’ = 0.

A line representing tx = to of euclidean K would be horizontal and on the X, X’ and Ksi axes. If, then, light really passed differently moving systems at the same physical velocity, the esynched clocks of k would be truly synchronous and the t’x’ = t’o line would be horizontal too. The angle of the t’x’ = t’o line isn’t due to the relative velocity of k. It’s due to the offsets physically put into the k-clocks by the act of esynching them. That, in turn, is due to the fact that light physically passes differently moving systems at different relative speeds, which are a measured constant only after the moving clocks have been esynched.

In Figure 5, event p is at tau = t = 1, ksi = 3, x = ksi – vtau = 2.4. Via the LTE it’s at x’ = 3. It isn’t at t’x’ = 0 of the intersecting-line t’x’ = t’o = 0 or, via projection, at t’ = .8 of line t’x’ = .8. It’s at t’ = ß(tau – vksi/c2) = -1.

For any given values of p(tau,ksi) the k-clock that will later register t’ = t’o plays no role in the value of t’p at that point. Neither, then, do geometrical projections from event p to or from the “rotated” line t’x’ = t’o. The clock at point x’ on the horizontal X’ axis registers the t’ value for any given point p, and it’s plotted by the k-clock at event p, which is a different event with different K’, K or k co-ordinates than event q! Indeed, the slanted t’x’ = t’o line, miscalled “the X’ axis”, plays no role in transformation equations that give us the cs k co-ordinates for a given event as plotted by systems K or K’.

It might now be asked why the t’x’ = t’o line isn’t at 59… degrees, as in Figure C. It’s because that line joins the points reached by successive k-clocks on X’, each of which is further to the right on an x-axis line than the prior clock. A line joining points reached by a given clock would be at a steeper angle. If we take cs K as “the stationary system” even though it’s inertially moving, then cs k is at rest to it. Even so, if we now plot the t’x’ = t’o = 0 line on an x,t graph, the clock at x’ = 1, which is at x = .8, would take .6/.8 = .75 seconds to register t’ = 0, thus the line would be at an angle atan(.8/.75) = 46.85 degrees.

The standard rotation diagrams not only rotate the “t’ axis” (which is actually only the positions of the origin clock) clockwise, they also rotate X’ counter-clockwise by the same degree. As shown above, however, since the X,X’ axes remain orthogonal to the t,t’ axes, if we rotate the t’ axis clockwise we also have to similarly rotate the X, X’ and t axes, while leaving the direction of v unchanged wrt the spacetime manifold. Though that might justify the angle in Figure 5, it violates Poincare’s “Lorentz Transformation Equations Group”, which includes circular transformations that place the direction of motion ON the thereafter coinciding Ksi,X,X’ axes of the various LTE systems. For that, we would have to rotate Ksi and tau as well and then redraw Figure 5; which we won’t, since rotations and geometric projections play no role in how the LTE and hyperbolic mathematics actually work nor in why the hyp-trig fits the LTE.

The spacetime terms “timelike” and “spacelike” and “light cones” and “here now” and “here elsewhere” and “rotation”, etc, are opaque window dressings that, together with so-called “geometrical projections”, block one from seeing the actual physical conditions that the LTE and the hyperbolic trignonometry itself rest on and impose. As we now look through that window we will let cs K replace K’ as the stationary euclidean system in the spacetime graphs.

Figure 6.

The hyberbolic curve in Figure 6 is included in standard Minkowski-rotation diagrams, though it’s meaning isn’t accurately explained. This one maps the velocity of an accelerating system, which we will let be cs k for the moment, as a function of the time t. It is moving to the left in the -x direction prior to t =0 but is accelerating to the right. At t = 0 it is at rest relative to system K. The instantaneous velocity of the accelerating system is given by v = at, in which a = d2x/dt2 = dv/dt. The shape of the curve is a function of the value of a. As t increases, the value of x (where the moving origin is per instant) increases exponentially. (That’s why it’s a hyperbolic curve.)

What is the value of x’ at t,x = 1,1? That depends on three things: (1) Does cs k remain euclidean? If so, the value is x’ = x-vt. If not, it depends on: (2) the value of phi(v) in the ETA, thus of l in Lorentz’s eqs 4 and 5. In order for the LTE to apply, the value must be phi(v) = l = 1, wherefore esynched cs k must deform by q,1,1,q; (3) the instantaneous value of v, which is a function the values of a and t in v = at.

Assuming that a = .1c/dt2, then at t = 1, v = (dv/dt)t = .1c. The hyp-trig then says,

x’ = cosh(r)x – sinh(r)t.

We know that x = t = 1, but we don’t yet know the value of r. How do we get that? From the hyperbolic curve of Figure 6! We draw a line through the given value of t and parallel to X that intersects the curve at a thereby specified point, here p. Then we draw a line tangent to p, measure its slope in radians and label that “r”. The value of hyperbolic slope r is a function of the value of v at point p at the time t. The instantaneous velocity of cs k, v = .1c, inserted into the hyp-trig equation r = atanh(v/c), then gives us the value of r at t = 1.

Since cs k is accelerating, however, the values of x, v and r change for every different value of t, so this value of r is restricted to t =1. How, then, can the hyp-trig now give us the time, t’, of an inertially moving system k for all values of x,y,z and t, as the LTE do? Answer: Once r has been found for a given value of v at a given value of t, the curve in Figure 6 has done its job and plays no further role. A heretofore invisible curve, of similar shape but with a completely different mathematical and physical meaning, now comes into play.

The Checkmate Curve.

Figure 7.

This hyperbolic curve isn’t plotting the instantaneous velocity of an accelerating system. It is just a graph of the value of r per system inertially moving at less than c. Though collectively equivalent to accelerating system k, each system now maintains its inertial velocity regardless of the values of t or x!

Once the value of r has been found, the purely abstract hyperbolic trig then gives us:

cosh(r) = ß = gamma = 1/q.

Lorentz’s eq 4 can therefore be written as x’ = ßlx = cosh(r)lx. Substituting x-vt for x, this becomes:

x’ = gamma(x-vt)l = cosh(x-vt)l.

The pure hyp-trig also gives us: sinh(r) = ßv, wherefore, if we let l = 1 this now becomes:

x’ = gamma(x – vt) = ßx – (ßv)t = cosh(r)x – sinh(r)t.

Eq 5 can be written as:

t’ = lt/ß – vßlx/c2 = qlt – vx’/c2 = t’o – sinh(r)lx/c2

which, though less revealing, is equivalent to Voigt’s 1887 local time equation, t’x’ = t’o – vx’/c2 in which it is clear that esynched clocks at different points x’ are offset by -vx’/c2 seconds compared to their origin clock. Again setting phi(v) = 1, substituting x – vt for x, letting gamma denote ß = 1/sqrt(1-v2/c2) and gathering terms we then reach,

t’ = t/ß – vß(x – vt) = gamma(t – vx) = [ß]t – [ßv]x = [cosh(r)]t – [sinh(r)]x = t’x’ = t’o – vx’/c2; in which it is implicit that the values of x and v , etc., are given as fractions of c = 1 unit-length/second.

The Minkowski mathematics isn’t really based on different “perspectives” due to “geometric projections” resulting from “rotations”. It’s based on hyperbolic trigonometry; in which the hyperbolic curve in the diagrams gives us the value of r, thus gamma and ßv, once v is known. Independently of any rotations or co-ordinate systems or spacetime diagrams at all, the hyp-trig equations substitute cosh(r) for gamma and sinh(r) for (gamma v), thus are just an equivalent way of writing the LTE per inertially moving system. They hold good if and only if each such system really does physically deform by q,1,1,q as a consequence of electromagnetic force changes due to its velocity through the local luminiferous substrate as referent.

Though verbally denied by the relativists, Lorentz’s physical deformations are built into the spacetime equations by the hyperbolic trigonometry itself! THAT is what Minkowski’s pure mathematics blocked out.

The mythical “rotations” and nonexistent geometrical projections are themselves the blindfold and so is everything Minkowski’s followers semantically fantasized.

Rather than having “explained” the Lorentz deformations, his procedure misled our physicists into thinking that there are no such physical deformations. As shown above, however, without them the LTE don’t hold good and with them Minkowski’s denial of the physical deformations on which the Lorentz Transformation Equations are based doesn’t hold good. Accordingly, other than as an abstract way to solve equations, the Theory of Relativity, as now taught in all our Universities the Minkowski way, is false.

Bibliography.

1. “Electromagnetic Phenomena in a System Moving With Any Velocity Less Than That

of Light” by H. A. Lorentz, 1904. Dover Books’ “The Principle of Relativity”, pages 9-34.

2. “Sur le Dynamique” by H. Poincare’, June, 1905.

3. “The Painted Pony” by du Gabriel, 1993.

4. “On the Electrodynamics of Moving Bodies” by A. Einstein, September, 1905. Dover

Books’, loc cit, pages 37-65.

5. “Space and Time” by H. Minkowski, 1908, with “Notes” by A Sommerfeld, 1908. Dover

Books, loc cit, pages 75-86.

6. “In the beginning there was God” by du Gabriel, 1994.